A ball is drawn at random from a box containing 10 balls numbered sequentially from 1 to 10. Let X be the number of the ball selected, let R be the event that X is an even number, let S be the event that $\displaystyle X\geq 6$, and let T be the event that $\displaystyle X\leq 4$. Which of the pairs (R,S), (R,T), and (S,T) are independent?

I can solve this mathematically as such:
$\displaystyle \newline P(R)=0.5\newline P(S)=0.5\newline P(T)=0.4$

Independence intuitively means that if you have information for one event that it doesn't effect the probability for another event.

So consider the two events: an even number and a number <= 4. If you know that something is even (2,4) then given that its also <= 4 you don't have any extra advantage of knowing what number it is. If its even then you have an equal chance of picking between 2 and 4 and if its odd again you have an equal chance of picking between 1 and 3.

No matter what the scenario, knowing one event doesn't help in predicting another and this is exactly what independence means intuitively in a probabilistic sense.

If one event did greatly effect the probability then you would get an P(A and B) != P(A)P(B).

The proof for independence is really simple. If B and A are independent then P(A|B) = P(A) and P(B|A) = P(B). Since P(A|B) = P(A and B)/P(B) = P(A) we multiply both sides by P(B) and get P(A|B) = P(A)*P(B) for independent events.